Convergence Rates of the Allen--Cahn Equation to Mean Curvature Flow: A Short Proof Based on Relative Entropies

DOI10.1137/20M1322182zbMath1454.53079arXiv2002.11994MaRDI QIDQ5141938

Julian Fischer, Tim Laux, Theresa M. Simon

Publication date: 29 December 2020

Published in: SIAM Journal on Mathematical Analysis (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/2002.11994

zbMATH Keywords

mean curvature flowreaction-diffusion equationsAllen-Cahn equationdiffuse interfacerelative entropy method

Mathematics Subject Classification ID

Singular perturbations in context of PDEs (35B25) Reaction-diffusion equations (35K57) Variational methods applied to PDEs (35A15) Calibrations and calibrated geometries (53C38) Flows related to mean curvature (53E10)

Related Items

Convergence of the scalar- and vector-valued Allen–Cahn equation to mean curvature flow with 90°-contact angle in higher dimensions, part I: Convergence result, A new energy stable fractional time stepping scheme for the Navier-Stokes/Allen-Cahn diffuse interface model, Convergence rates for the Allen-Cahn equation with boundary contact energy: the non-perturbative regime, The Sharp Interface Limit of a Navier–Stokes/Allen–Cahn System with Constant Mobility: Convergence Rates by a Relative Energy Approach, Analysis and approximations of an optimal control problem for the Allen-Cahn equation, Matrix-valued Allen-Cahn equation and the Keller-Rubinstein-Sternberg problem, Weak-strong uniqueness for volume-preserving mean curvature flow, Sharp Interface Limit for a Navier–Stokes/Allen–Cahn System with Different Viscosities, Analysis of a thermodynamically consistent Navier-Stokes-Cahn-Hilliard model, Nematic-isotropic phase transition in liquid crystals: a variational derivation of effective geometric motions, Convergence of the Allen--Cahn Equation with a Nonlinear Robin Boundary Condition to Mean Curvature Flow with Contact Angle Close to 90°

Cites Work

• On the motion of a curve by its binormal curvature
• Sharp interface limit for a Stokes/Allen-Cahn system
• The effect of a singular perturbation on nonconvex variational problems
• Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics
• Generation and propagation of interfaces for reaction-diffusion equations
• Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature
• Weak-strong uniqueness for the Navier-Stokes equation for two fluids with surface tension
• Phase transitions and generalized motion by mean curvature
• Convergence of the Allen‐Cahn Equation to Multiphase Mean Curvature Flow
• Geometrical Evolution of Developed Interfaces